Financial Economics
Fixed Income
Santiago Forte © ESADE
1
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
2
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
3
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Definition
– Fixed income securities are instruments issued either by
Governments (sovereign debt) or by companies (corporate debt)
whose distinguishing feature is that payments and maturity terms
are known in advance.
– The basic difference between sovereign debt and corporate debt
is that the default risk of sovereign debt can often be assumed to
be zero.
– The securities mentioned from here onwards are assumed to be
sovereign debt, unless otherwise specified.
4
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Basic components
– Frequency of payments: semi-annual, annual, etc.
– Amount of payments or cash flows: CF.
– Maturity: T.
0
1
2
CF1
CF2
  
T
CFT
5
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Fixed income securities: main types
– Regular bonds (plain vanilla): Periodic payment of a coupon c
plus the bond's face value (principal) N at maturity.
0
1
2
c
c
  
T
c
N
– Zero-coupon bonds: Single payment of face value at maturity.
0
  
T
N
6
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 1:
– 5-year Treasury bond with a face value of €100 and annual
coupon of 3%.
0
1
2
3
4
3
3
3
3
5
3
100
– Remark. Hereinafter, unless otherwise stated, face value assumed to be €100.
7
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
8
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Defining return
– Assume we invest in a bond currently worth V0 and we hold it
for T years. If we obtain a final value of VT in exchange, then the
annual return R produced by this investment over said period of
time is expressed by the following equation:
VT  V0  1  R 
T
V
R   T
 V0



1
T
1
9
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
10
Financial Economics
Fixed Income
Santiago Forte © ESADE
• In 2004, a brochure of the Spanish Treasury claimed:
Return
11
Financial Economics
Fixed Income
Santiago Forte © ESADE
• In 2004, a brochure of the Spanish Treasury claimed:
Return
Return (IRR)
12
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Further, it says:
13
Financial Economics
• Further, it says:
Fixed Income
Santiago Forte © ESADE
Return is guaranteed
and known …
14
Financial Economics
• Further, it says:
Fixed Income
Santiago Forte © ESADE
Return is guaranteed
and known …
… provided you keep
your investment until
maturity.
15
Financial Economics
Fixed Income
Santiago Forte © ESADE
• From previous information we should conclude
that:
– The return on sovereign debt is given by the IRR.
– The return on sovereign debt is guaranteed, provided
the investment is held until maturity.
• In general, this is not true!
16
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Let us recall the definition of the IRR of an investment
project:
“Discount rate which would make the project’s NPV equal to
zero”.
• So, what is the IRR of a bond?
The discount rate which would make the NPV of the “bond
purchase” project is equal to zero.
NPV   P 
c
c
c
N

 ... 

0
2
T
1  y  1  y 
1  y  1  y T
17
Financial Economics
Fixed Income
Santiago Forte © ESADE
• The IRR, the yield to maturity, or simply the yield as it is also known,
is therefore the discount rate that makes the market price of the
bond equal to the discounted value of its payments:
P
c
c
c
N

 ... 

2
T
1  y  1  y 
1  y  1  y T
• Usual interpretation: The bond's price is the value of its payments
discounted at the IRR.
• While this interpretation is useful in certain instances, it can be
misleading. No one values a Treasury bond by discounting its
payment at the IRR. It is the IRR that is deduced from the bond's
price.
• Note 1: How to value sovereign debt will be discussed later.
18
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Note 2: On the basis of what we already know about
financial mathematics:
T

1  y 1
N
P  c

T
T
y  1  y  1  y 
19
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 2: What is the IRR of a 5-year bond with a coupon of
4.25% if its market price is €99.13?
– If we equate the price and the discounted value of the payments
99.13  4.25 
1  y 5  1  100
5
5
y  1  y  1  y 
then
y  0.0445 ; 4.45%
i.e., the internal rate of return is 4.45%.
20
Financial Economics
Fixed Income
Santiago Forte © ESADE
• The Yield Curve shows the relationship between the IRR
of different Treasury bonds and their maturities.
• Example 3: The details of the bonds currently being
traded are:
T
c
P
Yield
1
3.00%
100.00
3.00%
2
5.00%
102.87
3.49%
3
6.00%
105.66
3.96%
4
5.25%
103.77
4.21%
5
4.25%
99.13
4.45%
21
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 3 (cont.): The graph of which is:
Yield Curve
5.00%
4.50%
4.00%
Yield
3.50%
3.00%
2.50%
2.00%
1.50%
1.00%
0.50%
0.00%
1
2
3
4
5
Maturity
22
Financial Economics
Fixed Income
Santiago Forte © ESADE
•
Example 4: You would like to make a safe, 5-year investment so that when
you finish your BBA and Master you'll be able to afford a short trip. The
brochure issued by the Spanish Treasury says that the “return” on its 5-year
bonds is 4.45% and that this is a “safe” investment, so you are thinking of
buying a few.
•
Questions:
a) How much should you receive in 5 years’ time if you invest €99.13 (the
current price of the bond) with an annual return of 4.45%?
b) Verify that you would receive that amount if you kept reinvesting the
coupons at that rate of interest (4.45%).
c) How much would you be paid if you could only reinvest the coupons at
3% p.a. (per annum)? In this case, what would the yearly return on your
original investment be?
23
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Conclusions:
– Treasury bonds guarantee the payment of fixed coupons in
addition to paying back the principal, but not the interest rate at
which we will be able to reinvest coupons.
– Hence, a bond’s IRR does not tell us what return we would
receive for holding our Treasury bonds until they mature and, in
general, it is not an investment with a safe return.
24
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 4 (cont.): The following table shows how the
value and final return of our investment vary according to
the reinvestment rate.
Reinvest. Rate
V5
Return
1.00%
121.68
4.18%
3.00%
122.56
4.34%
4.45%
123.23
4.45%
6.00%
123.96
4.57%
8.00%
124.93
4.74%
25
Financial Economics
Fixed Income
Santiago Forte © ESADE
• So now we know that realized returns are not (generally) guaranteed
even if the bonds are held until maturity, but we may wonder:
– What happens if we close our investment before maturity?
• Example 5: What would the final annual realized return be on a 5year bond if we sell it after 3 years? Assume:
a) That the reinvestment (and discount) rate is equal to the bond’s
original yield to maturity (IRR).
b) That the reinvestment (and discount) rate is 1%.
26
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 5 (cont.): The following table shows the return
obtained at different reinvestment and discount rates,
depending on the holding period (HP).
Reinv. & Disc. Rate
HP: 1 year
HP: 2 years
HP: 3 years
HP: 4 years
HP: 5 years
1.00%
17.96%
9.15%
6.36%
5.00%
4.18%
3.00%
9.85%
6.37%
5.24%
4.67%
4.34%
4.45%
4.45%
4.45%
4.45%
4.45%
4.45%
6.00%
-0.95%
2.46%
3.63%
4.22%
4.57%
8.00%
-7.36%
0.02%
2.61%
3.94%
4.74%
• Exercise 1: Calculate the final return (realized return) if the
bond is only held for 4 years and the reinvestment and
discount rate is 5%.
27
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Question
– So, can't we guarantee a certain return over a certain
period of time by investing in Treasury bonds?
• Answer
– Yes we can: By investing in zero-coupon bonds.
28
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 6:
– Let's imagine that besides issuing bonds with a coupon of 4.25%, the
government also issued 5-year zero-coupon bonds (with a face value of
€100). Let's also imagine that these bonds have a market value of
€80.25.
• Questions:
a) What is the IRR of these bonds?
b) If you buy one of these bonds, how much will you receive after 5 years
depending on the reinvestment and discount rate? What annual return
would you receive in the end if you hold the investment until maturity?
c) And what if the reinvestment and discount rate rise to 6% and you sell
the bonds after 2 years?
29
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 6 (cont.): Answers
– IRR: 4.50%
– Final annual return:
Reinv. & Disc. Rate
HP: 1 year
HP: 2 years
HP: 3 years
HP: 4 years
HP: 5 years
1.00%
19.75%
9.98%
6.90%
5.39%
4.50%
3.00%
10.72%
6.79%
5.51%
4.88%
4.50%
4.50%
4.50%
4.50%
4.50%
4.50%
4.50%
6.00%
-1.30%
2.29%
3.51%
4.13%
4.50%
8.00%
-8.41%
-0.54%
2.23%
3.64%
4.50%
30
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Conclusion: In the case of zero-coupon bonds
alone:
– The return on government bonds is measured by the
IRR.
– The return on government bonds is guaranteed,
provided that they are held until maturity.
31
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Therefore, if we want to lock in the return of our
investment over a given period of time, we must invest in
zero-coupon bonds that mature in that time.
• Are zero-coupon bonds with all maturities available?
• Main issues of Spanish Treasury bonds:
– Treasury Bills (Letras del Tesoro): 3, 6, 9 and 12-month zerocoupon bonds.
– Treasury Notes (Bonos del Estado): 3 and 5-year coupon bonds.
– Treasury Bonds (Obligaciones del Estado): 10 and 30-year
coupon bonds.
32
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Direct treasury issues only offer zero-coupon bonds that
mature in less than one year, but...
• … there are also securities known as strips. These
securities issued by financial institutions (sanctioned by
the Treasury) are coupon bonds that are segregated to
create a set of separate coupons and principals.
• These segregated securities are the equivalent of zerocoupon bonds issued by the Government.
33
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 7: Segregated 5-year bond
– A 5-year bond with a coupon of 4.25% can be regarded not as a
single bond with several payments, but as a portfolio of zerocoupon bonds:
Bond
T
N
1st
1
4.25
2nd
2
4.25
3rd
3
4.25
4th
4
4.25
5th
5
4.25
6th
5
100
34
Financial Economics
Fixed Income
Santiago Forte © ESADE
• So, if we want to lock in our return over a certain period
of time we should pop into our bank and buy a strip with
that maturity.
• Question:
What return should the bank offer us?
• The answer is in the Term Structure of Interest Rates:
TSIR.
35
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
36
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Definition
– The Term Structure of Interest Rates (TSIR), also
known as the zero-coupon yield curve, displays the
relationship between the IRR of (actual or theoretical)
government zero-coupon bonds and their respective
maturity. These values are known as spot rates.
37
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Comments
– Spot rates are usually unobservable, but can be
deducted from observable market information: The
trading price of (usually) coupon-bearing bonds.
– The method is known as Bootstrapping.
38
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 8: Let's take another look at the market
information about Treasury bonds:
T
c
P
1
3.00%
100.00
2
5.00%
102.87
3
6.00%
105.66
4
5.25%
103.77
5
4.25%
99.13
• What are the 1 to 5-year spot rates?
39
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Notation:
– r0t : t-years spot rate
– p0t : t-years discount factor, i.e., present value of the
Government's promise to pay €1 in t years.
p0 t 
1
1  r0t t
 1 

r0t  
p
 0t 
1
t
1
40
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Bootstrapping Method
– This method for estimating spot rates is based on regarding the bonds
being traded (with or without a coupon) as a portfolio of zero-coupon
bonds.
– From this viewpoint, the value of any bond may be expressed as :
P
c

c

c
1  r01  1  r02 2 1  r03 3
 ... 
cN
1  r0T T
which is the same as:
P  c  p01  c  p02  c  p03  ...  c  N  p0T
– These formulas can be used to calculate, sequentially, the series of spot
rates.
41
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 8 (cont.):
1.
r01
P1  103  p01
100  103  p01
p01  0.9709
1
1
 1 1
1 1
  1  
r01  
  1  0.0300 ; 3.00%
 0.9709 
 p01 
42
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 8 (cont.):
2.
r02
P2  5  p01  105  p02
102.87  5  0.9709  105  p02
p02  0.9335
 1 

r02  
 p02 
1
2
 1 
1  

 0.9335 
1
2
 1  0.0350 ; 3.50%
43
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 8 (cont.):
3.
r03
P3  6  p01  6  p02  106  p03
105.66  6  0.9709  6  0.9335  106  p03
p03  0.8900
1
1
 1 3
1 3
  1  
r03  
  1  0.0400 ; 4.00%
 0.8900 
 p03 
44
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 8 (cont.):
4.
r04
P4  5.25  p01  5.25  p02  5.25  p03  105.25  p04
103.77  5.25  0.9709  5.25  0.9335  5.25  0.8890  105.25  p04
p04  0.8466
1
1
 1  4
1  4
  1  
r04  
  1  0.0425 ; 4.25%
 0.8466 
 p04 
45
Financial Economics
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• Example 8 (cont.):
5.
r05
P5  4.25  p01  4.25  p02  4.25  p03  4.25  p04  104.25  p05
99.13  4.25  0.9709  4.25  0.9335  4.25  0.8890  4.25  0.8466  104.25  p05
p05  0.8025
1
1
 1 5
1 5
  1  
r05  
  1  0.0450 ; 4.50%
 0.8025 
 p05 
46
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 8 (cont.): Bootstrapping gives the following
zero-coupon yield curve:
TSIR
r ot
1
3.00%
2
3.50%
3
4.00%
4
4.25%
5
4.50%
Spot Rate
t
5.00%
4.50%
4.00%
3.50%
3.00%
2.50%
2.00%
1.50%
1.00%
0.50%
0.00%
1
2
3
4
5
Maturity
47
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Interpreting spot rates
– Spot rates reflect the real return implicitly demanded
by the market for lending the Government money,
depending on the contract term.
– They also reflect the real, guaranteed return we could
obtain by investing in fixed income securities,
depending on the contract term.
48
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Exercise 2: Three bonds are traded as follows:
T
c
P
1
7.00%
101.00
2
8.00%
106.00
3
9.00%
112.00
• Calculate the Term Structure of Interest Rates in this
instance.
49
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
50
Financial Economics
Fixed Income
Santiago Forte © ESADE
• One step further
– On the basis of information about the price of bonds (with or
without coupon) being traded, we have calculated the return
implicitly demanded by the market for lending the Government
money between the present day (0) and a future date (t): spot
rates.
– We can go one step further and deduct, on the basis of spot
rates, the return implicitly demanded by the market for lending
the Government money between two future moments in time (t’
and t): forward rates.
51
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Two equivalent investments
– Investment 1
• Lend the Government €1 from 0 to t, and obtain a return of
r0t
t
0
1 r0t t
1
– Investment 2
• Lend the Government €1 from 0 to t', where t’ < t, and obtain a return of
r0t '
• Reinvest the proceeds obtained in t’ until t and obtain a return (agreed
upon at 0) of f t ',t
0
t'
1
1  r0t ' t '
t
1  r0t ' t '  1  f t ',t t t '
52
Financial Economics
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• Both investments will be equivalent if and only if
1  r0t ' t '  1  f t ',t t t '  1  r0t t
that is, if and only if
1
f t ',t
 p  t t '
  0t '   1
 p0 t 
• f t ',t is the implicit rate between t’ and t, i.e., the forward
rate.
53
Financial Economics
Fixed Income
Santiago Forte © ESADE
• Example 9: Here are the discount factors we calculated
earlier:
t
pot
1
0.9709
2
0.9335
3
0.8900
4
0.8466
5
0.8025
• What is the implicit return of the Government debt
between year 1 and year 2? And between year 3 and
year 5?
54
Financial Economics
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• Expectation Theory:
– Forward rates not only represent the return implicitly demanded
by the market for lending the Government money between two
future moments in time, but also the market’s expectation on the
future evolution of spot rates:
E0 rt ',t   f t ',t
55
Financial Economics
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• Argument:
t'
t t '
t
– If E0 rt ',t   f t ',t , then 1  r0t '   1  E0 rt ',t   1  r0t  : The return from
lending the Government money between 0 and t will be lower than the
expected return from lending it money between 0 and t’, and reinvest the
procedes until t. Hence, nobody will be willing to lend the Government
money between 0 and t at the rate r0 t , that is, r0 t cannot be the
equilibrium rate between 0 and t .
 
t'
t t '
t
– If E0 rt ',t   f t ',t , then 1  r0t '   1  E0 rt ',t   1  r0t  : The cost (return)
assumed by the Government for borrowing money between 0 and t will be
higher that the expected cost of borrowing money between 0 and t’, and
refinance the debt until t. Hence, the Government will not be willing to
 
borrow between 0 y t at the rate r0 t , that is, r0 t cannot be the equilibrium
rate between 0 and t.
• Criticism: The Expectation Theory assumes risk-neutral investors.
56
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
57
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• Example 10: Imagine that the Treasury announces the
issue of a new 3-year bond with a coupon of 2% p.a.
(face value €100).
0
1
2
3
2
2
102
• How much do you think the Government will obtain from
the issue of each of these securities? Bear in mind the
details given in Example 8.
58
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• Bond pricing
– The value of new issues is determined on the basis of discount
factors. They tell us what is the market price of the Government's
promise to pay €1 in the future at each possible maturity.
P  c  p01  c  p02  c  p03  ...  c  N  p0T
– Remember that this is the same as considering a coupon bond
to be a portfolio of zero-coupon bonds, each valued by
discounting its payment by the respective spot rate.
P
c
c
c
cN


 ... 
2
3
1  r01  1  r02  1  r03 
1  r0T T
59
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• Example 10 (cont.):
– The discount factors we calculated earlier were:
t
p ot
1
0.9709
2
0.9335
3
0.8900
4
0.8466
5
0.8025
– Therefore, the price of the new bond will be:
P3 (c  2)  2  p01  2  p02  102  p03
 2  0.9709  2  0.9335  102  0.8900  94.49
60
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• Example 10 (cont.): What will the IRR of the new bond be?
Compare the result with the IRR of the 3-year bond already being
traded, and with the IRR of a (theoretical) 3-year zero-coupon bond.
– When calculating the IRR of the new bond, we estimate y such
that
94.49  2 
1  y 3  1  100
3
3
y  1  y  1  y 
y  0.0399 ; 3.99%
– Comparing the IRR of all 3-year bonds:
coupon
IRR
6
3.96%
2
3.99%
0
4.00%
61
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• Conclusions
– A bond's IRR depends not only on its maturity, but
also on its payment schedule (higher or lower
coupon).
– This is why the yield curve can never be used to
determine a bond's value: The yield to maturity of a
T-year bond cannot be used to discount the payments
of another T-year bond.
– To determine a bond's value we must use the TSIR.
62
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• Exercise 3:
– Calculate the price of a 4-year bond with a coupon of
1.5% p.a. and a face value of €1,000. Take into
account the details given in Example 8.
63
Financial Economics
Fixed Income
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SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
64
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• Remember that the value of any bond is determined by the value of
its payments discounted using spot rates:
P
c
c
c
cN


 ... 
2
3
1  r01  1  r02  1  r03 
1  r0T T
• Comments
– The risk of the bond is related to the risk of unforeseeable
changes in its market value.
– We know what the bond's face value, coupon and maturity are.
– Therefore, the only source of risk is possible changes in interest
rates.
65
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P
Fixed Income
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c
c
c
cN


 ... 
2
3
1  r01  1  r02  1  r03 
1  r0T T
• Mathematically, we can check that an increase in any spot rate will
cause the bond's market value to fall.
• Economic interpretation: Because the payments are fixed, if the
return demanded by the market for financing the Government
increases (i.e. the spot rates climb), the only way our bond can offer
said return to a potential buyer is if its selling price is reduced.
• Our aim: To determine how sensitive the bond's price is to
movements in interest rates.
66
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• Example 11: The following tables show the market information
again:
T
c
P
Yield
t
rot
pot
1
3.00%
100.00
3.00%
1
3.00%
0.9709
2
5.00%
102.87
3.49%
2
3.50%
0.9335
3
6.00%
105.66
3.96%
3
4.00%
0.8900
4
5.25%
103.77
4.21%
4
4.25%
0.8466
5
4.25%
99.13
4.45%
5
4.50%
0.8025
• Let's consider:
– A: The impact of a 1% increase in the 1-year spot rate together
with a fall of 0.5% in the 4-year spot rate.
– B: The impact of a 1% increase in all spot rates.
67
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• Example 11 (cont.): A. Increase of 1% in the 1-year spot rate; fall of
0.5% in the 4-year spot rate.
t
rot
pot
 rot
rot '
pot '
1
3.00%
0.9709
1.00%
4.00%
0.9615
2
3.50%
0.9335
0.00%
3.50%
0.9335
3
4.00%
0.8900
0.00%
4.00%
0.8890
4
4.25%
0.8466
-0.50%
3.75%
0.8631
5
4.50%
0.8025
0.00%
4.50%
0.8025
Yield '
 Yield
99.04
4.00%
1.00%
102.83
3.51%
0.02%
105.60
3.98%
0.02%
4.21%
105.45
3.76%
-0.45%
4.45%
99.16
4.44%
-0.01%
T
c
P
Yield
1
3.00%
100.00
3.00%
2
5.00%
102.87
3.49%
3
6.00%
105.66
3.96%
4
5.25%
103.77
5
4.25%
99.13
P'
68
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• Example 11 (cont.): B. An increase of 1% in all spot rates.
t
rot
pot
 rot
rot '
pot '
1
3.00%
0.9709
1.00%
4.00%
0.9615
2
3.50%
0.9335
1.00%
4.50%
0.9157
3
4.00%
0.8900
1.00%
5.00%
0.8638
4
4.25%
0.8466
1.00%
5.25%
0.8149
5
4.50%
0.8025
1.00%
5.50%
0.7651
Yield '
 Yield
99.04
4.00%
1.00%
100.96
4.49%
1.00%
102.83
4.96%
1.00%
4.21%
100.16
5.20%
1.00%
4.45%
94.88
5.45%
1.00%
T
c
P
Yield
1
3.00%
100.00
3.00%
2
5.00%
102.87
3.49%
3
6.00%
105.66
3.96%
4
5.25%
103.77
5
4.25%
99.13
P'
69
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• Conclusion and implications
– In practice, quantifying any possible change in spot rates would be too
complicated.
– Solution: Bear in mind that if, and only if, all spot rates changed by the
same amount (parallel change in the TSIR), then the change in the IRR
of any bond would tally with said variation.
– Remember the price/IRR relationship:
P
c
c
c N

 ... 
2
1  y  1  y 
1  y T
– Assuming a change in the IRR we could at least be able to quantify the
impact that an identical change in all spot types would have on the
bond's price.
70
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• Price vs interest rates:
P
y
• This relationship is:
– Inverse
– Non-linear
71
Financial Economics
Fixed Income
Santiago Forte © ESADE
SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
72
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Fixed Income
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• Example 12: How sensitive is the price of these bonds to
movements in interest rates?
T
c
P
Yield
1
3.00%
100.00
3.00%
2
5.00%
102.87
3.49%
3
6.00%
105.66
3.96%
4
5.25%
103.77
4.21%
5
4.25%
99.13
4.45%
• Duration gives an approximate answer to this question in a single
value.
• Duration is obtained by applying Taylor's approximation to changes
in the value of a function.
73
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• Taylor's Approximation
– If we have a function:
g  gy
– Taylor's n order approximation says that the total change in
in response to a change in y is:
g  g '  y y 
g
1
1
2
n
g ' '  y y   ...  g n  y y   errorn
2!
n!
– The higher the order
n
in the equation, the lower the error.
74
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• In our case:
P
T
CFt
c
c
cN


...



T
2
1  y  1  y 
1  y  t 1 1  y t
T
P y   
t 1
CFt
1  y t
• Considering Taylor's first-order approximation for changes in the
bond price in response to changes in interest rates:
P  P '  y  y  error1
P  P '  y   y
75
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• This may be shown as:
P
P
P0 , y0 
error1
P '  y   y
y
P'  y 
P y 
y
76
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• As a percentage:
Fixed Income
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P
1
 P '  y    y
P
P
• First derivative of P  y  :
P'  y   
T
CFt
1
 t 
1  y  t 1 1  y t
T
CFt
1
1
P

t 
  y
t
1  y  t 1 1  y  P
P
Duration (D)
Modified Duration (MD)
77
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• Hence, duration is the weighted average of the periods in which
payments take place.
• The weight of each period is the discounted value of the cash flow
generated in that period (discounted using the bond's yield to
maturity) over the bond's total value:
T
D  t 
t 1
CFt
1

t
1  y  P
Period
Weight
• Modified duration is simply:
MD 
D
1  y 
78
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• In short, duration and modified duration sum up in a single value the
sensitivity of a bond's price to interest rate movements.
• More specifically, they approximate the percentage variation in a
bond's price in response to a 1% change in interest rates.
D
MD
P
  D  y
P
P
  MD  y
P
• Modified duration is, however, a better (more exact) approximation.
79
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• Example 12 (cont.): How would a 1% increase in interest
rates affect the price of these bonds?
T
c
P
Yield
1
3.00%
100.00
3.00%
2
5.00%
102.87
3.49%
3
6.00%
105.66
3.96%
4
5.25%
103.77
4.21%
5
4.25%
99.13
4.45%
• Calculate the actual change and the approximation given
by duration and modified duration.
80
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• Example 12 (cont.): Let's calculate duration and modified
duration for the 3-years maturity bond.
Maturity
Coupon
IRR
3
6.0%
3.96%
Time in years
1
2
3
(1) Cash Flow
6
6
106
5.77
5.55
94.34
5.77
11.10
283.01
(2) Discounted Cash Flow (IRR)
(3) Price
105.66
(4) t x Discounted Cash Flow
(5) Duration
2.84
(6) Modified Duration
2.73
81
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• Example 12 (cont.): Or directly
T
D  t 
t 1
D
CFt
1

t
1  y  P

1
6
6
106 
 1
 2
 3
 2.84
2
105.66  1.0396 
1.0396
1.03963 
MD 
D
2.84

 2.73
1  y  1.0396
82
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• Example 12 (cont.): In the case of a 3-yr bond:
– Actual change:
P ( y  4.96%)  6 
1.04963  1  100  102.83
3
3
0.0496  1.0496  1.0496 
P
102.83  105.66
(real ) 
 0.0268 ;  2.68%
P
105.66
– Change forecast with duration:
P
( D)   D  y  2.84  0.01  0.0284 ;  2.84%
P
– Change forecast with modified duration:
P
( MD)   MD  y  2.73  0.01  0.0273 ;  2.73%
P
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• Example 12 (cont.): In the case of other bonds:
T
D
MD
y
Actual price
change
D forecast
MD forecast
1
1.00
0.97
1%
-0.96%
-1.00%
-0.97%
2
1.95
1.89
1%
-1.86%
-1.95%
-1.89%
3
2.84
2.73
1%
-2.68%
-2.84%
-2.73%
4
3.72
3.57
1%
-3.48%
-3.72%
-3.57%
5
4.61
4.41
1%
-4.29%
-4.61%
-4.41%
• In fact …
– … both duration and modified duration approximate the
percentage change in the price of a bond in response to a 1%
change in interest rates: a measure of the sensitivity of the
bond's price to interest rate movements.
– … modified duration gives a better approximation.
84
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• As shown in this graph, the smaller the change in interest rates, the
better should be the approximation.
P
P
error1
P0 , y0 
P '  y   y
y
P'  y 
P y 
y
85
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• Example 13: What impact would a 0.01% increase in interest rates
have on bond prices? In the case of a 3-yr bond:
– Actual change:
P ( y  3.97%)  6 
1.03973  1  100  105.63
3
3
0.0397  1.0397  1.0397 
P
105.63  105.66
(real ) 
 0.000273;  0.0273%
P
105.66
– Change forecast with duration:
P
( D)   D  y  2.84  0.0001  0.000284 ;  0.0284%
P
– Change forecast with modified duration:
P
( MD)   MD  y  2.73  0.0001  0.000273 ;  0.0273%
P
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• Example 13 (cont.): For the rest of the bonds:
T
y
Actual price change
D forecast
MD forecast
1
0.01%
-0.0097%
-0.0100%
-0.0097%
2
0.01%
-0.0189%
-0.0195%
-0.0189%
3
0.01%
-0.0273%
-0.0284%
-0.0273%
4
0.01%
-0.0357%
-0.0372%
-0.0357%
5
0.01%
-0.0441%
-0.0461%
-0.0441%
• Modified duration provide a very good approximation if
the change in interest rates is small.
87
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• Exercise 4: You have a 2-yr bond with a 3.5% coupon. The bond's
current IRR is 7%. You are concerned about the possibility of a
general increase of 2% in interest rates.
a) How would such an increase actually affect the bond's price?
b) What forecast would duration and modified duration give?
c) Modified duration always gives a better forecast than duration,
but, in this instance, is the forecast provided by modified duration
good? State your reasons.
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SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
89
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• Considering Taylor's second-order approximation for
changes in the bond price in response to interest rates
movements …
P  P '  y   y 
1
2
P ' '  y  y   error2
2
P  P '  y  y 
1
2
P' '  y  y 
2
• … the error will be smaller:
error2  error1
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• In percentages:
P
1
1
1
2
 P'  y   y   P' '  y   y 
2
P
P
P
Convexity (CX)
• To calculate convexity, we need the first derivative of P '  y  :
P' '  y  
T
CFt
1
  t  t  1
2
1  y  t 1
1  y t
• Resulting in:
CX 
T
CFt
1
1
t  t  1



t
2
1  y  P
1  y  t 1
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• Summary
– Taylor's second-order approximation for changes in the bond
price in response to interest rates movements says:
P
1
2
  MD  y   CX  y 
P
2
where
MD 
CX 
T
CFt
1
1
t 

t
1  y  t 1 1  y  P
T
CFt
1
1
t  t  1



t
2
1  y  P
1  y  t 1
92
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• Example 14: Example 12 asked what impact a 1%
increase in interest rates would have on the price of
these bonds:
T
c
P
Yield
1
3.00%
100.00
3.00%
2
5.00%
102.87
3.49%
3
6.00%
105.66
3.96%
4
5.25%
103.77
4.21%
5
4.25%
99.13
4.45%
• We calculated the actual change, and the approximation
given by duration and modified duration.
93
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• Example 14 (cont.): The results were:
T
D
MD
y
Actual price
change
D forecast
MD forecast
1
1.00
0.97
1%
-0.96%
-1.00%
-0.97%
2
1.95
1.89
1%
-1.86%
-1.95%
-1.89%
3
2.84
2.73
1%
-2.68%
-2.84%
-2.73%
4
3.72
3.57
1%
-3.48%
-3.72%
-3.57%
5
4.61
4.41
1%
-4.29%
-4.61%
-4.41%
• Let's see how our forecast improves by using Taylor's
second-order approximation: modified duration and
convexity.
94
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• Example 14 (cont.): Let's calculate the convexity of the 3-yr bond.
Maturity
Coupon
IRR
3
6.0%
3.96%
Time in years
(1) Cash Flow
(2) Discounted Cash Flow (IRR)
(3) Price
(4) t x Discounted Cash Flow
(5) Duration
(6) Modified Duration
(7) t x (t+1) x Discounted Cash Flow
(8) Convexity
1
2
3
6
5.77
6
5.55
106
94.34
5.77
11.10
283.01
11.54
33.31
1,132.05
105.66
2.84
2.73
10.31
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• Example 14 (cont.): Or directly:
CX 
CX 
T
CFt
1
1
t  t  1



t
2
1  y  P
1  y  t 1

1
1
6
6
106 

 2 
 6
 12 
 10.31
2
2
1.0396 105.66  1.0396
1.0396
1.03963 
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• Example 14 (cont.): In the case of the 3-yr bond:
– Actual change: -2.68%
– Change forecast by duration: -2.84%
– Change forecast by modified duration: -2.73%
– Change forecast by modified duration and convexity:
P
1
2
( MD, CX )   MD  y   CX  y 
P
2
1
P
2
( MD, CX )  2.73  0.01   10.31 0.01  0.0268 ;  2.68%
2
P
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• Example 14 (cont.): For the rest of the bonds:
T
D
MD
CX
y
Actual price
change
D forecast
MD forecast
MD and CX
forecast
1
1.00
0.97
1.89
1%
-0.96%
-1.00%
-0.97%
-0.96%
2
1.95
1.89
5.43
1%
-1.86%
-1.95%
-1.89%
-1.86%
3
2.84
2.73
10.31
1%
-2.68%
-2.84%
-2.73%
-2.68%
4
3.72
3.57
16.68
1%
-3.48%
-3.72%
-3.57%
-3.48%
5
4.61
4.41
24.63
1%
-4.29%
-4.61%
-4.41%
-4.29%
• Therefore …
– … combining modified duration and convexity gives a very good
approximation, even if interest rates change significantly.
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• Exercise 5:
– Calculate the convexity of the bond in Exercise 4. Remember
that we assumed a 2% increase in interest rates. What change
would be expected in the bond's price if we used modified
duration together with convexity? Compare with the actual
change.
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• So far we have seen that:
– Duration and modified duration sum up in a single value the
sensitivity of a bond's price to fluctuations in interest rates. This
is their information content.
– Both metrics are obtained from Taylor's first-order
approximation. They express specifically the expected
percentage change in a bond's price in response to a 1% change
in interest rates.
– Convexity arises from considering Taylor's second-order
approximation, enabling a more accurate prediction of how the
bond's price would change in response to a specific change in
interest rates.
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• However …
– … there is not much point using modified duration and convexity
to estimate the impact of a specific change in interest rates on a
bond's price. All we should to do is simply recalculate the price.
– So, what is the real information content of convexity?
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• Let's take another look at the price vs interest rates graph.
P
y
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• Modified duration estimated this relationship at a given
point by applying the first derivative:
P
P0 , y0 
y
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• Convexity provides additional information. Suppose that we
have two investments: A and B. Which has convexity most
pronounced? Which is best?
P
P0 , y0 
B
A
y
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• No matter how interest rates change, the final value of B
will always be higher than A: convexity effect.
P
P0 , y0 
B
A
y
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• Example 15: You hold a bond with a modified duration of
2.93.
a) According to modified duration, what is the approximate
impact of
- a 1.5% increase in interest rates?
- a 1.5% decrease in interest rates?
a) What would the answer to question (a) be if you also
knew that the bond's convexity was 9.31?
b) What would the answer be if its convexity was 13.85?
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• Example 15 (cont.): Identical effect?
a)
P
( MD; y  1.5%)  2.93  0.015  0.0440 ;  4.40%
P
P
( MD; y  1.5%)  2.93  (0.015)  0.0440 ; 4.40%
P
b)
P
1
2
( MD, CX ; y  1.5%)  2.93  0.015   9.31 0.015  0.0429 ;  4.29%
P
2
P
1
2
( MD, CX ; y  1.5%)  2.93   0.015   9.31  0.015  0.0450 ; 4.50%
P
2
c)
P
1
2
( MD, CX ; y  1.5%)  2.93  0.015  13.85  0.015  0.0424 ;  4.24%
P
2
P
1
2
( MD, CX ; y  1.5%)  2.93   0.015  13.85   0.015  0.0455 ; 4.55%
P
2
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• Example 15 (cont.): Convexity softens falls …
a)
P
( MD; y  1.5%)  2.93  0.015  0.0440 ;  4.40%
P
P
( MD; y  1.5%)  2.93  (0.015)  0.0440 ; 4.40%
P
b)
P
1
2
( MD, CX ; y  1.5%)  2.93  0.015   9.31 0.015  0.0429 ;  4.29%
P
2
P
1
2
( MD, CX ; y  1.5%)  2.93   0.015   9.31  0.015  0.0450 ; 4.50%
P
2
c)
P
1
2
( MD, CX ; y  1.5%)  2.93  0.015  13.85  0.015  0.0424 ;  4.24%
P
2
P
1
2
( MD, CX ; y  1.5%)  2.93   0.015  13.85   0.015  0.0455 ; 4.55%
P
2
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• Example 15 (cont.): … and makes increases larger.
a)
P
( MD; y  1.5%)  2.93  0.015  0.0440 ;  4.40%
P
P
( MD; y  1.5%)  2.93  (0.015)  0.0440 ; 4.40%
P
b)
P
1
2
( MD, CX ; y  1.5%)  2.93  0.015   9.31 0.015  0.0429 ;  4.29%
P
2
P
1
2
( MD, CX ; y  1.5%)  2.93   0.015   9.31  0.015  0.0450 ; 4.50%
P
2
c)
P
1
2
( MD, CX ; y  1.5%)  2.93  0.015  13.85  0.015  0.0424 ;  4.24%
P
2
P
1
2
( MD, CX ; y  1.5%)  2.93   0.015  13.85   0.015  0.0455 ; 4.55%
P
2
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• Conclusion: Modified duration and convexity provide
complementary information:
– Modified duration (also duration) shows how sensitive a bond's
price is to changes in interest rates:
• The greater the modified duration, the greater the sensitivity.
– Convexity complements this information by showing the extent to
which this sensitivity would lead to “more favorable changes”, or
“less unfavorable changes”, in the bond's price:
• The greater the convexity, the greater the price increases and
the smaller the decreases.
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SYLLABUS
1. Introduction
2. Return on fixed income securities
2.1. Internal rate of return (IRR): The Yield Curve
2.2. Term structure of interest rates: TSIR
2.3. Forward rates
3. Bond pricing
4. Risk metrics
4.1. Duration
4.2. Convexity
5. Corporate bonds
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• Corporate bonds are bonds issued by companies.
• A basic characteristic of corporate bonds is the inherent
risk of default. Whereas sovereign debt can be
considered (in certain circumstances) to have no risk of
default, the same cannot be said of corporate bonds.
• One immediate outcome is that the market expects the
return on corporate bonds to be higher than on Treasury
bonds: credit spread.
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• Empirical evidence:
Security
Average
Annual Return
Risk Premium
on Treasury
Bonds
Treasury Bonds
(US)
3.9%
0%
Corporate Bonds
6.0%
2.1%
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• Example 16:
– Nueva Rufasa has just issued 2-yr bonds with a face
value of €50,000 and 10% coupon.
– The market thinks that Nueva Rufasa has a high risk
of default, so it would require these bonds a credit
spread of 4% for 1-yr payments and of 5% for 2-yr
payments.
– What would the market value of each of these bonds
be?
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• Example 16 (cont.):
– As we saw, the 1 and 2-yr spot rates were:
t
r ot
1
3.00%
2
3.50%
– If these bonds were issued by the Treasury they would be worth:
PT 
c
c N


5,000

55,000
1  r01  1  r02 2 1.0300 1.03502
 56,195.89
– But because they are issued by Nueva Rufasa, their value is:
PNR 
c

cN

5,000

55,000
1  r01  CS01,NR  1  r02  CS02,NR 2 1.0700 1.08502
 51,391.58
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• The question is, how are credit spreads calculated?
• The credit spreads that the market requires for holding corporate
bonds depend on:
– The risk of default.
– The market's “appetite" for risk.
• One key factor in determining the risk of default is the credit rating.
• Rating agencies (Moody’s, Standard & Poor’s, Fitch) analyze the
bonds and evaluate their risk of default, and then assign a credit
rating.
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• A risk rating system
(RRS) is a system for
Santiago Forte © ESADE
An obligation rated ‘AAA’ has the highest rating assigned by Standard & Poor’s. The
obligor’s capacity to meet its financial commitment on the obligation is extremely
strong.
An obligation rated ‘AA’ differs from the highest rated obligations only to a small
degree. The obligor’s capacity to meet its financial commitment on the obligation is
very strong.
classifying assets
An obligation rated ‘A’ is somewhat more susceptible to the adverse effects of changes
in circumstances and economic conditions than obligations in higher rated categories.
However, the obligor’s capacity to meet its financial commitment on the obligation is
still strong.
according to their
An obligation rated ‘BBB’ exhibits adequate protection parameters. However, adverse
economic conditions or changing circumstances are more likely to lead to a weakened
capacity of the obligor to meet its financial commitment on the obligation.
Credit Risk level.
An obligation rated ‘BB’ is less vulnerable to non-payment than other speculative
issues. However, it faces major ongoing uncertainties or exposure to adverse business,
financial, or economic conditions that could lead to the obligor’s inadequate capacity to
meet its financial commitment on the obligation.
• S&P’s ratings.
An obligation rated ‘B’ is more vulnerable to non-payment than obligations rated ‘BB’,
but the obligor currently has the capacity to meet its financial commitment on the
obligation. Adverse business, financial, or economic conditions will likely impair the
obligor’s capacity or willingness to meet its financial commitment on the obligation.
An obligation rated ‘CCC’ is currently vulnerable to non-payment, and is dependent
upon favourable business, financial, and economic conditions for the obligor to meet its
financial commitment on the obligation. In the event of adverse business, financial, or
economic conditions, the obligor is not likely to have the capacity to meet its financial
commitment on the obligation.
An obligation rated ‘CC’ is currently highly vulnerable to non-payment.
The ‘C’ rating may be used to cover a situation where a bankruptcy petition has been
filed or similar action has been taken but payments on this obligation are being
continued. ‘C’ is also used for a preferred stock that is in arrears (as well as for junior
debt of issuers rated ‘CCC-’ and ‘CC’).
The ‘D’ rating, unlike other ratings, is not prospective; rather, it is used only where a
default has actually occurred—and not where a default is only expected.
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• The three steps in assigning a credit rating:
• Step 1. Set the company’s initial rating on the basis of its financial
data.
• Step 2. Set the company’s final rating, analyzing possible
downgrades from the initial rating as a function of:
– 2.a) The sector’s situation and the company’s relative position
– 2.b) The capacity of executives and other qualitative data
– 2.c) The quality of the financial information available
– 2.d) Country risk
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• The three steps in assigning a credit rating:
• Step 3. Set the rating of the specific asset by studying possible
adjustments to the company’s overall rating taking into account:
– 3.a) Underwriting by third parties
– 3.b) Maturity
– 3.c) Collateral
– 3.d) Other clauses
–
.
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• From credit ratings to default probabilities
– Assuming that the risk rating system has objective
criteria and is systematic, we may expect:
• 1. Assets to have a lower probability of nonpayment, the higher their rating is.
• 2. Assets with the same rating to have a similar
probability of non-payment.
– The question is: What are these probabilities?
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• From credit ratings to default probabilities
• Procedure: Let’s consider the historic rate of default for
each rating. This rate will be the default probability of the
companies that now have this rating.
– Problem: The number of observations that even the largest
banks have is insufficient to generate reliable statistics.
– Importance of the rating agencies: The major agencies
(Moody’s, Standard and Poor’s) have about 100 years’
experience and thousands of default observations (3,600 in the
case of Moody’s).
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• From credit ratings to default probabilities ...
– Default probability (1 year) as a function of S&P ratings
Rating
Default Rate
AAA
0.00%
AA
0.00%
A
0.06%
BBB
0.18%
BB
1.06%
B
5.20%
CCC
19.79%
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• ... and also ...
– Migration probabilities (1 year): S&P
End rating
AAA
AA
A
BBB
BB
B
CCC
Default
AAA
90.81
8.33
0.68
0.06
0.12
0.00
0.00
0.00
AA
0.70
90.65
7.79
0.64
0.06
0.14
0.02
0.00
A
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
BBB
0.02
0.33
5.95
86.93
5.30
1.17
0.12
0.18
BB
0.03
0.14
0.67
7.73
80.53
8.84
1.00
1.06
B
0.00
0.11
0.24
0.43
6.48
83.46
4.07
5.20
CCC
0.22
0.00
0.22
1.30
2.38
11.24
64.86
19.79
Initial rating
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• Credit spreads of zero-coupon bonds according to rating and
maturity.
(US October 2001 figures in basis points: 100 bp = 1%. Source: Chris Marrison, “The Fundamentals of Risk
Measurement”)
Rating\Maturity
1
2
3
5
7
10
30
AAA
38
43
48
62
72
81
92
AA
48
58
63
77
92
101
112
A
73
83
103
117
137
156
165
BBB
118
133
148
162
182
201
220
BB
275
300
325
350
375
450
575
B
500
550
600
675
725
775
950
CCC
700
750
900
1,000
1,100
1,250
1,500
• The worse the rating (and the longer the maturity), the higher the
credit spread.
124
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• Exercise 6: Imagine that S&P gives Nueva Rufasa's
issue a CCC rating. Considering the figures in the table
above, what is the value of these bonds?
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•
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Exercise 7 (2009-2010 mid-term exam):
Treasury bonds can often be assumed to have no risk of default. In short, the bonds issued in local currency by
states with sovereignty over their monetary policies can be considered to have no default risk: to meet their obligations, all
they have to do is printing more banknotes! In the case of a monetary union, however, things are not so simple. The member
states can issue bonds in the common currency, but do not own the press that prints money.
Imagine a monetary union between two states: Allemania and Grettia. They both issue 1 and 2-year bonds in
“Allegrettos”, their common currency. Allemania has a robust economy and the market feels that although it has no
sovereignty over its monetary policy, there is no risk of it defaulting on its debt. Grettia's situation is more complicated
because its overspending in the past makes the market think it has a considerable risk of default.
We have the following information about the bonds issued by the two states (face value €100):
T
1
2
Allemania
c
P
2.50%
100.49
4.50%
102.43
Grettia
c
3.80%
6.25%
P
97.01
90.54
a) What credit spreads would Grettia have to accept in comparison with Allemania when issuing 1-year zerocoupon bonds?; and when issuing 2-year zero-coupon bonds?
b) Allemania and Grettia now make a new issue of new 2-year bonds with a face value of €1,000. Imagine that both
states issue their respective bonds at par (market value of both bonds: €1,000). What difference would there be between the
coupons paid by Grettia and Allemania? Without needing any further calculations, what IRR would these new bonds have at
the time of issue?
126
127